Find the value of K for which the function F given as f(x)={1-cos x/2x2, if x=0 is continuous at x=0 K ifx=0

Question: Find the value of K for which the function F given as f(x)={1-cos x/2x2, if x=0 is continuous at x=0 K ifx=0

The correct answer is -

To find the value of K for which the function f(x) is continuous at x=0, we need to evaluate the limit of f(x) as x approaches 0 and compare it with the value of f(0).

First, let's substitute x=0 into the given function:

f(0) = (1 - cos(0)) / (2*0^2) = 0/0, which is an indeterminate form.

To evaluate the limit of f(x) as x approaches 0, we can use L'Hopital's rule:

lim x→0 [1 - cos(x)] / (2x^2) = lim x→0 [sin(x)] / (4x) = lim x→0 [cos(x)] / 4 = 1/4

Therefore, to make f(x) continuous at x=0, we need to choose K such that:

K = 1/4

So, the value of K for which the function f(x) is continuous at x=0 is 1/4.